Morphism

Abstract: Introduction. 1: Weighted Sobolev spaces. 2: Capacity. 3: Supersolutions and the obstacle problem. 4: Refined Sobolev spaces. 5: Variational integrals. 6: A-harmonic functions. 7: A superharmonic functions. 8: Balayage. 9: Perron's method, barriers, and resolutivity. 10: Polar sets. 11: A-harmonic measure. 12: Fine topology. 13: Harmonic morphisms. 14: Quasiregular mappings. 15: Ap-weights and Jacobians of quasiconformal mappings. 16: Axiomatic nonlinear potential theory. Appendix I: The existence of solutions. Appendix II: The John-Nirenberg lemma. Bibliography. List of symbols. Index

Abstract: This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kinds of manifolds (algebraic, analytic, etc.). Sheaves also appear in logic as carriers for models of set theory as well as for the semantics of other types of logic. Grothendieck introduced a topos as a category of sheaves for algebraic geometry. Subsequently, Lawvere and Tierney obtained elementary axioms for such (more general) categories. This introduction to topos theory begins with a number of illustrative examples that explain the origin of these ideas and then describes the sheafification process and the properties of an elementary topos. The applications to axiomatic set theory and the use in forcing (the Independence of the Continuum Hypothesis and of the Axiom of Choice) are then described. Geometric morphisms- like continuous maps of spaces and the construction of classifying topoi, for example those related to local rings and simplicial sets, next appear, followed by the use of locales (pointless spaces) and the construction of topoi related to geometric languages and logic. This is the first text to address all of these varied aspects of topos theory at the graduate student level.

Abstract: LetK be the structure got by forgetting the composition law of morphisms in a given category. A linear representation ofK is given by a map V associating with any morphism ϕ: a→e ofK a linear vector space map V(ϕ): V(a)→V(e). We classify thoseK having only finitely many isomorphy classes of indecomposable linear representations. This classification is related to an old paper by Yoshii [3].

Abstract: A1 Regular and Cartesian Closed Categories A2 Toposes - Basic Theory A3 Allegories A4 Geometric Morphisms - Basic Theory B1 Fibrations and Indexed Categories B2 Internal and Locally Internal Categories B3 Toposes over a base B4 BTop/S as a 2-Category